how to square a number in your head (idea)

Ok, first you have to know how to square a binomial. That's a fairly simple process: Just multiply the first terms the middle terms, twice, then the outer terms and sum them up.

(x + 1)2 = x2 + 2x + 1

(x + 2)2 = x2 + 4x + 4

(x + 3)2 = x2 + 6x + 9

(x + 4)2 = x2 + 8x + 16

(x - 1)2 = x2 - 2x + 1

(x - 2)2 = x2 - 4x + 4

(x - 3)2 = x2 - 6x + 9

(x - 4)2 = x2 - 8x + 16

Ok, now, I'm sure you're able to square any number from 1 to 10, so squaring 20, 30, 40, 50, 60 etc. shouldn't be a problem because you can just tack on two extra zeros at the end. 302 = 900. You get the idea.

Ok, now suppose you want to square 32. You know how to square 30, which we'll call x, and you know how to square (x + 2). So since x is 30, and (x + 2)2 is x2 + 4x + 4, 32 squared must be 900 + 120 + 4, which is 1024.

Another interesting trick is squaring any number that ends in 5, like 35, 45, 55, 65. Lets use 65. You just multiply the first number, that is the number in the tens column, which is 6, multiply it by the number after it, which is 7, and just tack on 25 to the result. So 652 is 4225.

Personally I hate squaringbinomials that involve subtraction. I just want to add, no subtraction involved. So what I do is, use the previous trick to square any number. Say I want to square 77. So based on my first trick you would use (x - 3) squared, where x is 80. But if you combine it with the previous trick you won't have to subtract. You just do (x + 2) squared where x is 75. So x2 + 4x + 4 = 5625 + 300 + 4 = 5929.

This might seem hard the first couple of times you do it, but if you practice a little bit you'll be able to square any number less than (and I suppose equal to) 100 easily.